Two Circles and a Limit
Proof #3 by Mariano Perez de la Cruz

A stationary circle of radius $3$ is centered at $(3, 0).$ Another circle of variable radius $r$ is centered at the origin and meets the positive $y-axis$ in point $A.$ Let $B$ be the common point of the two circles in the upper half-plane. Let $E$ be the intersection of $AB,$ extended, with the $x-axis.$ What happens to $E$ as $r$ grows smaller and smaller?

Let $F$ be the midpoint of $AB,$ $D$ the intersection of $OF$ with the stationary circle in the upper half-plane, $C$ the rightmost point of the stationary circle.

$D$ is the circumcenter of the isosceles $\Delta AOB.$ This is because angles $BOD$ and $BCD$ are inscribed angles subtended by the same arc, whereas $\angle AOF = \angle BOF,$ on one hand and $\angle AOF = \angle OCD,$ on the other. (The latter because the angles have orthogonal sides.) It follows that $OD = DB.$ In the limit, when $A,$ $B,$ $F$ tend to coalesce, $D$ will tend to the midpoint of $OF.$

$\Delta ODC$ is right as is the angle $OFE.$ Therefore, $EA\parallel CD.$ With the previous observation, the ratio $EF/CD$ will tend to $2,$ while $CD$ will approach $OC.$ $EF$ will then tend to twice the segment $OC.$